(In)approximability of maximum minimal FVS

Abstract

We study the approximability of the NP-complete Maximum Minimal Feedback Vertex Set problem. Informally, this natural problem seems to lie in an intermediate space between two more well-studied problems of this type: Maximum Minimal Vertex Cover, for which the best achievable approximation ratio is √ n, and Upper Dominating Set, which does not admit any n 1− approximation. We confirm and quantify this intuition by showing the first non-trivial polynomial time approximation for Max Min FVS with a ratio of O(n 2/3 ), as well as a matching hardness of approximation bound of n 2/3− , improving the previous known hardness of n 1/2− . Along the way, we also obtain an O(∆)-approximation and show that this is asymptotically best possible, and we improve the bound for which the problem is NP-hard from ∆ ≥ 9 to ∆ ≥ 6.Having settled the problem's approximability in polynomial time, we move to the context of super-polynomial time. We devise a generalization of our approximation algorithm which, for any desired approximation ratio r, produces an r-approximate solution in time n O(n/r 3/2 ) . This time-approximation trade-off is essentially tight: we show that under the ETH, for any ratio r and > 0, no algorithm can r-approximate this problem in time n O((n/r 3/2 ) 1− ) , hence we precisely characterize the approximability of the problem for the whole spectrum between polynomial and sub-exponential time, up to an arbitrarily small constant in the second exponent.

Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation

“…Studying Max-Min and Min-Max versions of some famous optimization problems is not a new idea, and it has recently attracted some interest in the literature: Minimum Maximal Independent Set [6,15,19] (also known as Minimum Independent Dominating Set), Maximum Minimal Vertex Cover [5,26], Maximum Minimal Separator [16], Maximum Minimal Cut [12], Minimum Maximal Knapsack [1,13,14] (also known as Lazy Bureaucrat Problem), Maximum Minimal Feedback Vertex Set [11]. In fact, the original motivation for studying these problems was to analyze the performance of naive heuristics compared to the natural Max and Min versions, but these Max-Min and Min-Max problems have gradually revealed some surprising combinatorial structures, which makes them as interesting as their natural Max and Min versions.…”

Parameterized Max Min Feedback Vertex Set

“…This was subsequently improved to NP-completeness for graphs of maximum degree 6 by Dublois et al [19], who also present an approximation algorithm with ratio n 2/3 and proved that this is optimal unless P=NP. A consequence of the polynomial time approximation algorithm of [19] was the existence of a kernel of order O(k 3 ), which implied that the problem is fixed-parameter tractable with respect to the natural parameter k. Some evidence that this kernel size may be optimal was later given by [2]. We note also that the problem can easily be seen to be FPT parameterized by treewidth (indeed even by clique-width) as the property that a set is a minimal feedback vertex set is MSO 1 -expressible, so standard algorithmic meta-theorems apply.…”

“…Max Min FVS was first shown to be NP-complete even on graphs of maximum degree 9 by Mishra and Sikdar [31]. This was subsequently improved to NP-completeness for graphs of maximum degree 6 by Dublois et al [19], who also present an approximation algorithm with ratio n 2/3 and proved that this is optimal unless P=NP. A consequence of the polynomial time approximation algorithm of [19] was the existence of a kernel of order O(k 3 ), which implied that the problem is fixed-parameter tractable with respect to the natural parameter k. Some evidence that this kernel size may be optimal was later given by [2].…”